The Nikkor 18-55 has minimum focusing distance of 0.9 feet (0.28 m); if you really want to get closer, you'll need a macro lens. There are very few options in this price range. There's the Nikkor AF-S 40mm f/2.8. Other options, such as a used Nikkor 60mm f/2.8 Micro or a new Sigma 50mm f/2.8 Macro, would leave you with manual focus (which is actually quite okay for close-up shots).
However, if you meant that you want to magnify your subjects more without actually getting closer to them, these focal lengths will obviously be too short (you already have those). If you had asked for image quality and low light performance, I would have suggested the Rokinon 85mm f/1.4, which is praised for those properties, but its rather heavy weight (1.4 lb / 635 g) and lack of AF will probably put you off. I don't know any other primes worth considering in your price class (the Nifty Fifty is cheaper and not longer than your kit zoom).
So in this case, you're left with zooms. On Nikon, I have only used 70-300 VR which was quite impressive for its focusing speed and image quality, and it seems to be available $380 used. Perhaps your budget can be stretched a little or some hunting around will find a better deal.